Integral de $$$e^{x y}$$$ em relação a $$$x$$$

Encontre $$$\int e^{x y}\, dx$$$.

Seja $$$u=x y$$$.

Então $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{y}$$$.

Portanto,

$${\color{red}{\int{e^{x y} d x}}} = {\color{red}{\int{\frac{e^{u}}{y} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{y}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{e^{u}}{y} d u}}} = {\color{red}{\frac{\int{e^{u} d u}}{y}}}$$

A integral da função exponencial é $$$\int{e^{u} d u} = e^{u}$$$:

$$\frac{{\color{red}{\int{e^{u} d u}}}}{y} = \frac{{\color{red}{e^{u}}}}{y}$$

Recorde que $$$u=x y$$$:

$$\frac{e^{{\color{red}{u}}}}{y} = \frac{e^{{\color{red}{x y}}}}{y}$$

Portanto,

$$\int{e^{x y} d x} = \frac{e^{x y}}{y}$$

Adicione a constante de integração:

$$\int{e^{x y} d x} = \frac{e^{x y}}{y}+C$$

$$$\int e^{x y}\, dx = \frac{e^{x y}}{y} + C$$$A